Topological quantum field theory and the Gromov-Witten theory of
curves in Calabi-Yau threefolds -- Jim Bryan, March 5, 2004

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Topological quantum field theory, as formulated by Atiyah, has
provided a general framework for understanding invariants of manifolds. 
The structure of TQFTs in dimension 1+1 (i.e. surfaces with boundary) is
completely understood by elementary means -- yet they can still yield 
surprising results. For each positive integer d, we define a one-parameter 
family of (1+1)-dimensional TQFTs Z_d(t) which specializes at t=0 to the 
famous Witten-Dijkgraaf-Freed-Quinn TQFT for gauge theory with gauge 
group the dth symmetric group. Our family of TQFTs completely encodes 
all the degree d local Gromov-Witten invariants of a curve (of arbitrary 
genus) in a Calabi-Yau threefold. This provides us with a "structure 
theorem" for these local invariants (a.k.a. multiple cover formulas). Using 
these ideas we completely determine the local invariants for d < 6.